Csc[z] == (Sqrt[-z^2]/(z Sqrt[Cosh[I z]^2 - 1])) (-1)^Floor[1/2 + Re[z]/Pi] (1 - (1 + (-1)^(Floor[-(1/2) - Re[z]/Pi] + Floor[1/2 + Re[z]/Pi])) UnitStep[-Im[z]]) /; !Element[(2 z - Pi)/(2 Pi), Integers]

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mtext> </mtext> </mrow> <mi> z </mi> </mfrac> <mo> &#8290; </mo> <mfrac> <mrow> <mn> 1 </mn> <mtext> </mtext> </mrow> <msqrt> <mrow> <mrow> <msup> <mi> cosh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> &#8970; </mo> <mrow> <mfrac> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> &#960; </mi> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> &#8970; </mo> <mrow> <mfrac> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> &#960; </mi> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> <mo> + </mo> <mrow> <mo> &#8970; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> &#960; </mi> </mfrac> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mtext> &#8203; </mtext> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mi> &#960; </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <csc /> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <cosh /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <plus /> <apply> <times /> <apply> <real /> <ci> z </ci> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <floor /> <apply> <plus /> <apply> <times /> <apply> <real /> <ci> z </ci> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <floor /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <real /> <ci> z </ci> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> UnitStep </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <imaginary /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <notin /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29